The mechanical oscillators, also called regulators, of timepieces are composed of a flywheel, called a balance wheel, and a spiral spring, called a hairspring, which is fixed, on the one hand, to the balance wheel staff and, on the other hand, to a pallet bridge in which the balance wheel staff pivots. The balance wheel/hairspring oscillates about its equilibrium position at a frequency that must be kept as constant as possible, as it determines the operation of the timepiece. For a homogeneous and uniform hairspring, the period of oscillation of such oscillators is given by the expression:
  T  =      2    ⁢    π    ⁢                                        J            b                    ·                      L            s                                                E            s                    ·                      I            s                              in which:                Jb is the total moment of inertia of the balance wheel/hairspring;        Ls represents the active length of the hairspring;        Es is the elastic modulus of the hairspring; and        Is is the second moment of section of the hairspring.        
A temperature variation results in a variation in the oscillation period such that, to the first order:
            Δ      ⁢                          ⁢      T        T    =            1      2        ⁢          {                                    Δ            ⁢                                                  ⁢                          J              b                                            J            b                          +                              Δ            ⁢                                                  ⁢                          L              s                                            L            s                          -                              Δ            ⁢                                                  ⁢                          E              s                                            E            s                          -                              Δ            ⁢                                                  ⁢                          I              s                                            I            s                              }      i.e. an expansion effect on Jb, Ls and Is and a thermoelasticity effect on Es. With an increase in temperature, the first three terms are generally positive (expansion of the balance wheel, elongation of the hairspring and reduction in Young's modules) and bring about a loss, whereas the last term is negative (increase in the cross section of the hairspring) and brings about a gain.
In the past, several methods for compensating for the temperature drift of the frequency have been proposed in order to alleviate this problem. Mention may in particular be made of methods of compensation by thermal modification of the moment of inertia of the balance wheel (for example a bimetallic balance wheel made of steel and brass) or by the use of a special alloy (for example invar) for hairsprings having a very low thermoelastic coefficient. These methods remain complicated, difficult to implement and consequently expensive.
More recently, in its European patent application EP 02026147.5 the Applicant described a method for the thermal compensation of the spring constant of a spiral spring, consisting in thermally oxidizing a hairspring produced in a silicon substrate. In the case of hairsprings made of steel of the invar type (for example the house alloy Nivarox-Far S.A.), spiral springs made of oxidized silicon make it possible to regulate the thermal behavior of the spring itself, possibly with a slight overcompensation by a few ppm/° C. This overcompensation limitation is due to the maximum oxide thickness that can be produced in practice (currently less than 4 μm) and to the minimum tolerable width of the cross section of the silicon hairspring (greater than 40 μm). Consequently, the balance wheel must also be thermally compensated. This can be obtained, for example, using an alloy of the “glucydur” type (a copper-beryllium alloy, also called “glucinium”) or else other alloys having a very low thermal expansion coefficient. This method is also complicated and, no more than the other more conventional methods, does not make it possible to correct for other isochronism defects, such as those due for example to various frictional effects in the oscillator, to the balance wheel being out of balance, to the center of mass of the hairspring being off-center, etc.